Measure and fractal sets

Speaker: Ahsan Ali, Montana State University, USA

Abstract: This is a short reading course aimed to introduce students to modern real analysis

(also called measure theory) and use it to study fractal sets. Roughly speaking, fractal sets

are subsets of Rn that can have non-integer (even irrational) dimension. However, the notion

of ’dimension’ will need to be rigorously defined for non-integer dimension values to make

sense.

Prerequisites: Basic Real Analysis (Sequences, Series, Convergence, Limsup, Liminf, Sequence of Functions, Pointwise Convergence, Uniform Convergence and Riemann Integration).

Plan: (Tentative)

1. What is a Measure?

Description: The basic concepts of σ-algebras, measurable sets, measurable functions

and measure will be defined. Many classical examples of measures will be given and

emphasis will be given on the definition of the Lebesgue Measure.

2. What is Integration?

Description: First, the class of (measurable) simple functions will be defined. Then,

some classical convergence theorems like Fatou’s Lemma and Monotone Convergence

Theorem will be discussed using the class of non-negative functions. Lastly, the class

of integrable functions will be defined and used to discuss the Lebesgue Dominated

Convergence Theorem. Also, a brief contrast of the between Lebesgue and Riemann

Integration will be given.

3. What is a Fractal Set?

Description: A detailed introduction and motivation to study fractal sets will be

given along with some classical examples. Then, the intricate notion of dimension will

be analysed and defined to allow non-integer dimension values. Using this definition,

some explicit computations will be made to determine dimension. To conclude, some

applications and references will be mentioned for further study.

Primary Textbooks:

1. The Elements of Integration and Lebesgue Measure by R.G. Bartle

2.  Real Analysis: Measure Theory, Integration & Hilbert Spaces by Elias M. Stein and Rami Shakarchi

3.  Fractal Geometry: Mathematical Foundations and Applications by Kenneth Falconer

Reference Textbooks:

1. Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland

2. Fractals in Probability and Analysis by Christopher J. Bishop and Yuval Peres